# Infinitely many hyperbolic Coxeter groups through dimension 19

@article{Allcock2006InfinitelyMH, title={Infinitely many hyperbolic Coxeter groups through dimension 19}, author={Daniel Allcock}, journal={Geometry \& Topology}, year={2006}, volume={10}, pages={737-758} }

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinct Coxeter groups in H^n with noncompact fundamental domain of volume less than or equal to V grows at least exponentially with respect to V. The same result holds for cocompact…

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## References

SHOWING 1-10 OF 26 REFERENCES

### Hyperbolic Coxeter N-Polytopes with n+2 Facets

- Mathematics
- 2003

In this paper, we classify all the hyperbolic noncompact Coxeter polytopes of finite volume whose combinatorial type is either that of a pyramid over a product of two simplices or a product of two…

### On right-angled reflection groups in hyperbolic spaces

- Mathematics
- 2005

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space $\Bbb H^n$ may only exist if $n\leq 14.$ We also provide a family of such polyhedra of dimensions…

### Compact Hyperbolic Coxeter n-Polytopes with n+3 Facets

- MathematicsElectron. J. Comb.
- 2007

We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, $4\le n\le 7$.…

### Leech roots and Vinberg groups

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1982

We study the recently defined Leech roots, which have many remarkable properties. They are the fundamental roots for the even unimodular lattice in Lorentzian space R25,1, and correspond one for one…

### The classification of compact hyperbolic Coxeterd-polytopes withd+2 facets

- Mathematics
- 1996

This paper provides a list of all compact hyperbolic Coxeter polytopes the combinatorial type of which is the product of two simplices of dimension greater than 1. Combined with results of…

### Coxeter groups, Lorentzian lattices, and K 3 surfaces

- Mathematics
- 1998

The main result of this paper describes the normalizer of a finite parabolic subgroup of a (possibly infinite) Coxeter group. We use this to compute the automorphism groups of some Lorentzian…

### Sphere Packings, Lattices and Groups

- MathematicsGrundlehren der mathematischen Wissenschaften
- 1988

The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to…

### ON GROUPS OF UNIT ELEMENTS OF CERTAIN QUADRATIC FORMS

- Mathematics
- 1972

It is shown that, for the groups of unit elements of certain integral quadratic forms of signature (n, 1), there exist subgroups of finite index, generated by reflections; and the generators and…

### Sphere packings, I

- Physics, MathematicsDiscret. Comput. Geom.
- 1997

A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

### Discrete groups generated by reflections in the faces of symplicial prisms in Lobachevskian spaces

- Mathematics
- 1974

The paper provides the Coxeter schemes of all the discrete groups generated by reflections in the faces of simplicial prisms the combinatory type of which is the product of a simplex and a segment in…